Integration

himanshu121

Here is the Problem:

[tex] \int_0^\pi\theta^2cosec^2\theta d\theta[/tex]


I want to do this problem by parts and finding hard to find shortest way to do the problem any shortest way pls

mathman

The straightforward way is probably the easiest. integral of csc² is -ctn and integral of ctn is log(sin). Integrating log(sin) may take a little work.

PrudensOptimus

cos*sec^2(x) or cosx*sec^2(x)?

master_coda

Isn't it csc(x)?

himanshu121

It is cosecant(x) i.e csc(x)
This is probably the easiest way of doing which i too have tried but it is not the shortest way

There are many ways of doing a problem i am looking for shortest way
Thnxs

master_coda

How about:

[tex]
\begin{align*}
\int_0^\pi\theta^2\csc^2\theta\;d\theta
&=\int_0^1\theta^2\csc^2\theta\;d\theta+\int_1^\pi\theta^2\csc^2\theta\ ;d\theta \\
&>\int_0^1\theta^2\csc^2\theta\;d\theta+\int_1^\pi\csc^2\theta\;d\th eta \\
&=\int_0^1\theta^2\csc^2\theta\;d\theta+\left[-\cot\theta\right]_1^\pi
\end{align*}
[/tex]

Now since [itex]\lim\limits_{\theta\rightarrow\pi}(-\cot\theta)=+\infty[/itex] we know that the rightmost term diverges. Moreover, the left integral (the one from 0 to 1) is clearly positive. Thus the original integral clearly diverges.

jk7711

master_coda:
the integral of x^2*(cscx)^2 from 1 to pi comes out to be about 4.2
the integral of (cscx)^2 from 1 to pi goes to inifinite. I believe to say the integral of x^2*(cscx)^2 from 1 to pi is greater than the integral of (cscx)^2 from 1 to pi since according to the values it came out as then the formula should be switched around and divergence would not be proved.

himanshu121:
Best way to do it would be to plug it into a calculator or look it up in a table. Otherwise you're probably stuck with integrating log(sin(x)).

jk

himanshu121

We are not allowed to use calculator in India till we are undergraduate
I found the way but dont know whether it is shortest one or not but definetly i wont stuck at log(sinx)



[tex]

I = \int_0^\pi\theta^2\csc^2\theta d\theta [/tex]

[tex]I= \int_0^\pi\ (\pi-\theta)^2\csc^2\theta d\theta [/tex]

this gives

[tex] \pi\int_0^\pi\csc^2 \theta d\theta = 2\int_0^\pi\theta\csc^2\theta d\theta [/tex]

integrating by parts with one part as [tex]\theta[/tex] and other as [tex]\theta\csc^2\theta d\theta [/tex]

i will get[tex] \int_0^\pi \cot\theta d\theta = \log(csc\theta-cot\theta)
[/tex]

much easier than integrating log(sinx)

But another problem is how i will put the limits in cotx from 0 to pi in both cases it is infinity and i know there is no break in the function cotx b/w these points

Hurkyl

I agree with master_coda...

Where did you come up with this?! My calculator says 3*10^14 with the caveat of "questionable accuracy" (though, 3*10^14 is a good approximation of infinity. [;)])

chroot

Yes, for large values of 3*10^14.

- Warren

master_coda

I'm pretty sure that the integral diverges. My algebra, my calculator and my computer all agree.

Also, if it's any help, [itex]\csc\theta-\cot\theta=\frac{1-\cos\theta}{\sin\theta}[/itex].

jk7711

hurkyl. yeah that 4.2 was kinda off huh? just keeping you on your toes i guess:P i entered it in wrong but i redid it and came up with what you got.

i used u=(x^2)csc^2(x) and dv=dx and came up with the first term (x^3*csc^2(x)) going to infinite so i think you might be right coda.

jk

harsh

Why dont you try using the tabular method to doing this problem.
Since theta^2 will eventually go to zero if you keep on taking the derivatives, you should do it by tabular method